On extension of funtors
L. Karhevska, T. Radul
Abstrat.A.Chigogidze denedforeahnormalfuntorontheategoryComp
anextensionwhihisanormalfuntorontheategoryTyh. Weonsiderthis
extensionforanyfuntorontheategoryCompandinvestigatewhihproperties
itpreservesfromthedenitionofnormalfuntor. Weinvestigateaswellsome
topologialpropertiesofsuhextension.
Keywords:Chigogidzeextensionoffuntors,1-preimagepreservingproperty
Classiation: 18B30,54B30,57N20
Introdution
Thegeneraltheoryoffuntorsatingin theategoryCompofompatHaus-
dorspaes(ompata)andontinuousmappingswasfoundedbyE.V.Shhepin
[15℄. Hedistinguished some elementary properties of suh funtors and dened
thenotionofnormalfuntorthat hasbeomeveryfruitful. Thelassofnormal
funtorsinludesmanylassialonstrutions: thehyperspaeexp,thefuntorof
probabilitymeasuresP,thepowerfuntorandmanyotherfuntors(see[13℄,[9℄
formoredetails). Butsomeimportantfuntorsdonotsatisfysomeoftheprop-
ertiesfrom the Shhepin list. Omitting somepropertiesweobtainwider lasses
offuntorssuhasweaklynormalfuntorsandalmostnormalfuntors.
Thepropertiesfromthedenitionofnormalfuntorouldbeeasilygeneralized
forthefuntors ontheategoryTyhof Tyhonovspaesand ontinuousmaps.
Letusremarkthat TyhontainsCompasasubategory. A.Chigogidzedened
foreahnormalfuntorontheategoryCompanextensionwhihisanormalfun-
torontheategoryTyh[6℄. Thisextension ouldbeonsidered foranyfuntor
on theategoryComp. Butthe situation is moreompliated for wider lasses
offuntors. Forexample,theextensionoftheprojetivepowerfuntor(whihis
weakly normal)doesnotpreserveembeddings,whihmakessuh extensionuse-
less(seeforexample[13, p.67℄). However,ifweapplytheChigogidzeextension
tosuhweakly normalfuntorsasthefuntorO oforder-preservingfuntionals,
thefuntorGofinlusionhyperspaes,thesuperextension,weobtainfuntorson
theategoryTyhwhih preserveembeddings.
Themainaimofthis paperistoinvestigatewhih propertiesfrom thedeni-
tionof normalfuntor are preservedbyChigogidze extension, speially we on-
entrateourattentionon thepreserving of embeddings. Theresults devoted to
preserving property whih is ruial for preservingof embeddings. In Setion 3
weonsiderwhihfuntorshavethe1-preimagespreservingproperty.
T.BanakhandR. CautyobtainedtopologiallassiationoftheChigogidze
extensionofthefuntorofprobabilitymeasuresforseparablemetrispaes. We
generalizethisresultto onvexfuntorsinSetion4.
x1
All spaes are assumed to be Tyhonov, all mappings are ontinuous. All
funtors are assumed to be ovariant. In the present paper we will onsider
funtorsatingintwoategories: theategoryTyhanditssubategoryComp.
Letusreallthedenitionofnormalfuntor. A funtorF :Comp!Compis
alledmonomorphi (epimorphi)ifitpreservesembeddings(surjetions). Fora
monomorphifuntorF andanembeddingi:A!X weshallidentifythespae
F(A)andthesubspaeF(i)(F(A))F(X).
A monomorphi funtor F is said to be preimage-preserving iffor eah map
f :X !Y andeahlosedsubsetAY wehave(F(f)) 1
(F(A))=F(f 1
(A)).
ForamonomorphifuntorFtheintersetion-preservingpropertyisdenedas
follows: F( T
fX
j2Ag)= T
fF(X
)j2Ag foreveryfamilyfX
j2Ag
oflosedsubsetsofX.
A funtor F is alled ontinuous if it preservesthe limits of inverse systems
S =fX
;p
;Agoveradireted setA. Letusalsonote that foranyontinuous
funtor F : Comp ! Comp the map F : C(X;Y) ! C(FX;FY) (the spae
C(X;Y)isonsideredwiththeompat-opentopology)isontinuous.
Finally, afuntor F isalled weight-preserving ifw(X)=w(F(X))for every
inniteX2Comp.
A funtor F is alled normal [15℄ if it is ontinuous, monomorphi, epimor-
phi,preservesweight,intersetions,preimages, singletonsandtheemptyspae.
A funtor F is said to be weakly normal (almost normal) if it satises all the
properties fromthedenition of anormalfuntorexeptperhapsthepreimage-
preservingproperty(epimorphiity)(see[13℄ formoredetails).
Similarly, one andene the sameproperties fora funtor F : Tyh ! Tyh
withtheonlydierenethatthepropertyofpreservingsurjetionsisreplaedby
thepropertyofsendingk-overingmapsto surjetions(reallthat f :X !Y is
ak-overingmapifforanyompatsetB Y thereexistsaompatsetAX
withf(A)=B)(see[13, Denition2.7.1℄).
A. Chigogidze dened an extension onstrution of a funtor in Comp onto
Tyh thefollowingway[6℄. Forany normalfuntor F :Comp!Compand any
X 2Tyh,thespae
F
(X)=fa2F(X)j there existsaompatset AX with a2F(A)g
isonsideredwiththetopologyinduedfromF(X),whereXistheStone-
Ceh
ompatiationofthespaeX. Next,givenanyontinuousmappingf :X !Y
betweenTyhonovspaes,putF
(f)=F(f)j
F (X)
. ThenF
formsaovariant
funtor in the ategoryTyh. Chigogidzeshowedthat in ase F is normal, the
funtorF
isalso normal.
x2
LetusmodifytheChigogidzeonstrutionforanyfuntorF :Comp!Comp.
For X2Tyh weput
F
(X)=fa2F(X)j thereexistsaompatset AX
with a2F(i
A
)(F(A))g
where by i
A
wedenote the naturalembedding i
A
: A,!X (wedo notassume
that the map F(i
A
) is an embedding). Evidently F
preserves empty set and
one-pointspaeiF does.
Now weonsider the problem when F
preserves embeddings. Extension of
anynormalfuntorpreservesembeddings, but, ifwedropthepreimagepreserv-
ing property, the situation ould be dierent. However, theexamples from the
introdutionshowthatthepreimage-preservingpropertyisnotneessary. Wede-
nesomeweakerpropertywhihwillgiveusaneessaryandsuÆientondition.
Denition1. WesaythatamonomorphifuntorF :Comp!Comppreserves
1-preimages, ifforanyf :X !Y,where X;Y 2Comp, anylosed AY suh
thatfj
f 1
(A)
isahomeomorphism,wehavethat(Ff) 1
(FA)=F(f 1
(A)). (Let
us remark itis equivalent to theonditionthat the map Ff j(Ff) 1
(FA) is a
homeomorphism.)
Letus note that this denition wasindependentlyintrodued byT. Banakh,
M.KlymenkoandA.Kuharski[3℄.
Proposition1. If F isamonomorphifuntorthatpreserves1-preimagesinthe
lassofopenmappings,thenF preserves1-preimages.
Proof: Takeanymappingf : X !Y suh that fj
f 1
(A)
is ahomeomorphism
for somelosed subsetA Y. Let i
1
: X !XY be theembedding dened
bythe formulai
1
(x)=(x;f(x)). Denote Z =XY=", where therelation" is
givenby"=fpr 1
Y
(a)ja2Ag(pr
Y
:XY !Y istherespetiveprojetion).
Let q : XY ! Z be the quotient mapping. The map h : Z ! Y given by
the onditions h(z) = y for any z = (x;y) 2 Z nq(X A) and h(z) = a for
any z = q(pr 1
Y
(a)), a 2 A, is open and satises the following twoonditions:
pr
Y
=hÆq,hj
h 1
(A)
isahomeomorphism. Apparently, themapi=qÆi
1 isan
embedding, moreover, hÆi = f. Sine F preserves1-preimages in the lass of
openmappings,wehave(Fh) 1
(FA)=F(h 1
(A)),whih givesus theequality
(Ff) 1
(FA)=F(f 1
(A)).
Proposition2. If F isamonomorphifuntorthatpreserves1-preimages,then
F
preservesembeddings.
Proof: Takeanyembedding f :X !Y. Thenthemap F
(f)islosed asthe
restritionof a losedmap onto afull preimage and, moreover,injetive,hene
anembedding.
ForanyX 2TyhandanyitsompatiationbX weandene
F
b
(X)=fa2F(bX)j thereisaompatsubset AX
with a2F(A)gF(bX)
andonsideritwiththerespetivesubspaetopology.
Corollary 1. If F is a monomorphi, 1-preimage-preserving funtor, then
F
(X)
= F
b
(X)foranyTyhonovspaeX and itsompatiationbX.
Proposition 3. If F is monomorphi, preserves1-preimagesand weight, then
F
preservesweight.
Proof: Thestatementfollowsfromthepreviousorollaryandthefat thatfor
any X 2 Tyh there exists its ompatiation bX whih has the same weight
asX.
Asthefollowingpropositionshows,thereverseimpliationtothatofProposi-
tion2alsoholds.
Proposition 4. Let F be aontinuous funtorsuh that F
preservesembed-
dings. ThenF preserves1-preimages.
Proof: Assume the ontrary. Then there exist a map f : X ! Y and a
losed subsetAY suh that fj
f 1
(A)
is ahomeomorphism andFf 1
(FA) 6=
F(f 1
(A)). We ansuppose that the map f is open by Proposition 1. There
exist 2FAand2FXnF(f 1
(A))suhthatFf()=. Wewillonstruta
spaeS2TyhanditsompatiationSsuhthatthemapF
(id
S ):F
(S)!
F
(S)=F(S)isnotanembedding,whereid
S
:S!S isanidentityembed-
ding.
FirstputZ =XN,wherethespaeofnaturalnumbersNisonsideredwith
thedisrete topologyand N =N [fg isthe one-point ompatiationof N.
Deneaontinuousfuntiong:Z !Y byg(x;n)=f(x)foranyx2X,n2N.
LetT =Z =" beaquotientspae, where " isan equivalene relationdened by
its lasses of equivalene ffxg j x 2 (X nf 1
(A))Ng[fg 1
(y)\X fg j
y 2Y nAg[ffagN ja2f 1
(A)g. Byq:Z !T wedenotethe respetive
quotient mapping. Then the map h:T !Y dened by theequality g =hÆq
is ontinuous. Theset D = q(Xfg) is ompat asa ontinuousimage of a
ompat set and moreoverhj
D
is one-to-one, hene ahomeomorphism between
D andY. Wedenote byj:Y !T theinverse embedding. Also,foranyn2N
thespaeS
n
=q(Xfng)is homeomorphito X andwedenej
n
:X !T by
j
n
(x)=q(x;n). ThenwehavehÆj
n
=f. FinallynotethatTisaompatiation
1
Put
n
=F(j
n
)()forn2N. Thesequenej
n
onvergestojÆf in thespae
C(X;T). SineF isontinuous,thesequeneF(j
n
)onvergestoF(jÆf)inthe
spaeC(FX;FT). Henethesequene
n
onvergestoF(jÆf)()=F(j)()2
F(q(f 1
(A)N)) .
NowonsiderF
(S)asasubspaeofF(S). Deneamaps
1
:S!X bythe
onditions
1 Æj
n
=id
X
foralln. Letusshowtheontinuityofs
1
. Considerany
pointt2SandanyopenneighborhoodU ofs
1
(t)inX. Sinethemapf isopen,
theset q(UN) =q((UN)[(f 1
(f(U))fg)) isan openset in T whih
ontainsthepointt. The set V =q(UN)\S isan openneighborhood of t
suhthat s
1
(V)U.
Lets:S!X betheextensionofs
1
. ThenFs(
n
)=2=F(f 1
(A)). Then
thesequene
n
does notonvergeto anyelementof F(q(f 1
(A)N)). The
propositionisproved.
Propositions2and4yieldthefollowing
Theorem 1. Forany ontinuous monomorphi funtor F the funtor F
pre-
servesembeddingsifandonlyif F preserves1-preimages.
Theproofofthefollowingpropositionisaroutinehekingandweomitit.
Proposition 5. LetF :Comp!Compbeafuntor.
(1) If F preserves embeddings, 1-preimages and intersetions then F
pre-
servesintersetions.
(2) If F preservesembeddingsand preimagesthenF
preservespreimages.
(3) If F preservessurjetionsthenF
sendsk-overingmapsto surjetions.
Now let us onsider ontinuity of the Chigogidze extension. The following
example shows that in the absene of the preimage-preserving property of the
funtorF,itis diÆulttospeakofontinuityofF
, sineeventheextensionof
suhknownweaklynormalfuntorasGdoesnotpossessit.
Example. LetusdenetheinlusionhyperspaefuntorG. Reallthatalosed
subset A 2 exp 2
X (where X 2 Comp) is alled aninlusion hyperspae, if for
everyA2AandeveryB2expX theinlusionABimpliesB 2A. ThenGX
isthespaeofallinlusion hyperspaeswith theinduedtopologyfromexp 2
X.
For any map f : X ! Y dene Gf : GX ! GY by Gf(A) = fB 2 expY j
f(A)B forsomeA2Ag. ThefuntorG isweakly normal(see [13℄for more
details). InthenextsetionwewillseethatthefuntorGpreserves1-preimages.
Let us show that the funtor G
is not ontinuous. Consider the following
inverse system. Foranyn2N put X
n
=N f1;:::;ng(here thespaesN and
f1;:::;ng are onsidered with the disrete topology). Dene p m
n : X
m
! X
n ,
where m n, in the following way: p m
n
(x;k) = (x;min fk;ng). We obtained
theinversesystem S =fX
m
;p m
n
;Ng. Thenthe limitspae X =limS ishome-
omorphi to the spae N A (here A = N = N [fg is the one-point om-
patiation of N, i.e. a onvergent sequene; also we put to begreater than
any natural number), and the limit projetionsp : X ! X anbe given by
p
n
(x;k)=(x;min fk;ng),k 2N. Theontinuity ofG
meansthat limG
(p
n ):
G
(limS)!limG
(S)isahomeomorphism.HerebothG
(limS)andlimG
(S)
anbethoughtassubspaesofG(bX),wherebisaompatiationofXwiththe
propertybX =limS. Therstinlusion followsfrom Corollary1,andthese-
ondinlusionis duetoontinuityofG(heneG(limS)=G(bX)=limG(S))
and existeneof the embedding limG
(S),!limG(S) whih is thelimit of a
morphismthatnaturallyembedseahG
(X
n
)into G(X
n ).
Now we will onstrut K 2 limG
(S) whih does not belong to limG
(p
n )
(G
(limS)). Considerthe spaeX embedded into itsompatiation bX. For
anyn2AnfgputK
n
=f1;:::;ngfng. Ifwewanttoobtainalosedfamilyof
sets,thesetK
=Nfgmustbeaddedtothefamily e
K=fK
n g
n2N
. Nowput
K=fBbXjK
n
B forsomen2Ag. ThenK2limG
(S). However,there
isnoelementC2G
(limS)withlimG
(p
n
)(C)=K. Indeed,theprojetionof
anyompatsetBX ontothefatorN ofNA isnite,henelimG
(p
n )(C)
does not ontain K
or ontains some nite subsets in N N fg. Hene,
limG
(p
n
),beingnotsurjetive,isnotahomeomorphism.
x3
We start this setion with denitions of some funtors we deal with in this
paper. Let X be aompatum. By C(X) we denote the Banah spae of all
ontinuous funtions : X ! R with the usual sup-norm. We onsider C(X)
with thenaturalorder. Let : C(X) !R be afuntional (we donot suppose
a priori that is linear or ontinuous). We say that is 1) non-expanding if
j(') ( )jd('; )forall'; 2C(X);2)weakly additive ifforanyfuntion
2 C(X) and any 2R we have(+
X
) =()+ (by
X
wedenote the
onstant funtion); 3) preserves order if for any '; 2 C(X) suh that '
theinequality(') ( )holds; 4)linear if forany , 2R and forany two
funtions ,2C(X)wehave(+ )=()+( ).
NowforanyspaeX denoteVX= Q
'2C(X)
[min';max'℄. Foranymapping
f :X !Y denethemapVf asfollows: Vf()(')=('Æf)forevery2VX,
'2C(Y). ThenV isaovariantfuntorintheategoryComp[11℄.
LetusremarkthatthespaeVX ouldbeonsideredasthespaeofallfun-
tionals :C(X)!R withtheonlyonditionmin'(X)(')max'(X)for
every'2C(X). ByEX wedenotethesubsetofVX denedbytheondition1)
(non-expandingfuntionals;see[5℄formoredetails),byEAXthesubsetdened
by the onditions 1) and 2). The onditions 2) and 3) dene the subset OX
(order-preservingfuntionals,see[10℄);nally,theonditions3)and4)denethe
well-knownsubset PX (probability measures,see for example[13℄). Foramap
f :X !Y themappingFf, where F is oneofP, O, EA, E, isdened asthe
restrition of Vf on FX. It is easy to hek that the onstrutions P, O, EA
andE denesubfuntorsofV. ItisknownthatthefuntorsOandEareweakly
normal(see [10℄ and [5℄). Using the sameargumentsone anhek that EA is
The question arises naturally whih of the funtors dened above have the
propertyofpreserving1-preimages. Itiseasytohekthatwehavetheinlusions
PXOX EAXEXVX. WewillshowthatthefuntorEAsatisesthis
property and E doesnot. Sine subfuntors inherit the 1-preimagespreserving
property,thisistheompleteanswer. Letusalsoremark thattheresultsof[11℄
and[12℄showthatmanyotherknownfuntorsouldbeonsideredassubfuntors
of EA, for example the superextension, the hyperspae funtor, the inlusion
hyperspaefuntoret. Thisshowsthatthelassoffuntorswiththe1-preimages
preservingpropertyiswideenough.
Westartwithadenition ofanAR -ompatum. Reallthat aompatumX
isalledanabsoluteretrat(brieyX 2AR ) ifforanyembeddingi:X !Z of
X into ompatumZ theimagei(X)isaretratofZ.
Thenextlemmawill beneededinthefollowingdisussion.
Lemma1. LetF beamonomorphisubfuntor of V whihpreservesinterse-
tionsandBbealosedsubsetofaompatumX. Then2FBi ('
1 )=('
2 )
foreah'
1
;'
2
2C(X)suhthat'
1 j
B
='
2 j
B .
Proof: Neessity. Theinlusion2FB FXmeansthatthereexists
0 2FB
withF(i
B )(
0
)=,wherei
B
:B!X isthenaturalembedding. Hene,forany
'
1 ,'
2
2C(X)suhthat'
1 j
B
='
2 j
B
wehave('
1 )=
0 ('
1 Æi
B )=
0 ('
2 Æi
B )=
('
2 ).
SuÆieny. Wean ndanembedding j:B ,!Y,where Y 2AR . Dene Z
tobethequotientspaeofthedisjointunionX[Y obtainedbyattahingX and
Y byB. Denote byr:Z !Y aretrationmapping.
Now takeany 2 FX FZ with the property ('
1
)= ('
2
) for eah '
1 ,
'
2
2 C(X) suh that '
1 j
B
= '
2 j
B
. We laim that F(r)() =. Indeed, take
any'2C(Z). ThenF(r)()(')=('Ær)=(') sine'Ærj
Y
='j
Y
. Hene,
2FX\FY =FB.
Proposition 6. ThefuntorEApreserves1-preimages.
Proof: Letf :X !Y beaontinuousopenmapbetweenompata X andY
andB be alosedsubsetof Y suhthat fj
f 1
(B)
is ahomeomorphism. Choose
any 2 EA(B) EA(Y). Using Lemma 1we an dene
0
2 EA(f 1
(B))
by the ondition
0
(') = ( ) for eah ' 2 C(X) and 2 C(Y) suh that
Æf jf 1
(B)='j
f 1
(B) .
Itisenoughtoshowthatforeah2(EA(f)) 1
()wehave=
0
. Suppose
the ontrary. Then there exist ' 2 C(X) and 2 C(Y) suh that Æf j
f 1
(B)='j
f 1
(B)
and(')6=( ). Weansuppose that(')>( ). Dene
afuntion 0
:Y !R by 0
(y)=max'f 1
(y)foranyy 2Y. Thefuntion 0
isontinuoussinef isopen. Also,sine 0
j
B
= j
B
,wehavethat( 0
)=( ).
Put =( 0
D)Æf,whereD=supfmax'f 1
(y) min'f 1
(y)jy2Yg. Then
d(;') D but (') () =(') ((
0
D)Æf) = (') ( 0
)+D =
(') ( )+D>Dandweobtainaontradition. Theproofissimilarforthe
Hene,EApreserves1-preimagesinthelassofopenmappings,and,byPropo-
sition1,wearedone.
Proposition 7. The funtor of nonexpanding funtionals E does not preserve
1-preimages.
Proof: Considerthemappingf :X!Y betweendisretespaesX =fx;y;s;tg
andY =fa;b;gwhihisdenedasfollows: f(x)=a,f(y)=b,f(s)=f(t)=.
PutA=f'2C(X)j'(s) ='(t)g. Denethefuntional :A!R asfollows:
(') =minf'(x);'(y)g if 'j
fx;yg
0, (') = maxf'(x);'(y)g if 'j
fx;yg 0,
and (') =0otherwise. Onean hekthat isnonexpanding. Nowtakethe
funtion :X !R dened asfollows (x)=1, (y)= 1, (s)=0, (t)=4.
Oneanhekthatweanextend toanonexpandingfuntionalonA[f gby
deningitsvalueon tobe 1. Thisnewfuntionalanbefurther extendedto
anonexpanding funtional on thewhole C(X) [5℄. Denote this extension by e.
Evidently,Ef(e)2E(fa;bg). Ontheotherhand,e2=E(fx;yg).
x4
Weonsider in this setion a monomorphiontinuous funtor F whih pre-
servesintersetions,weight,emptyset,pointand1-preimages. Weinvestigatethe
topologyofthespaeF
Y whereY isametrizableseparablenon-ompatspae.
Weonsider Y asadense subsetofametrizableompatumX. Itfollowsfrom
Corollary1that F
Y ishomeomorphito F
b
Y FX (whereX isonsidered as
aompatiationbY ofY)andinwhatfollowsweidentifyF
Y withF
b
Y. Also,
thepropertiesweimposeonF implythatF
Y isadensepropersubspaeofFX.
T. Banakh proved in [1℄ that F
Y is a F
-subset of FX when Y is loally
ompat;F
Y isF
Æ
-subsetwhenY isaG
Æ
-subset. IfY isnotaG
Æ
-subset,then
F
Y isnotanalyti.
WeonsiderintheHilbertubeQ=[ 1;1℄
!
thefollowingsubsets: =f(t
i )2
Qjsup
i jt
i
j<1g;=f(t
i
)2Qjt
i
6=0fornitely manyigand
!
Q
!
= Q.
Itisshownin[2℄thatanyanalytiP
Y ishomeomorphitooneofthespaes
,or
!
. Wegeneralizethisresultforonvexfuntors.
ByConvwedenotetheategoryofonvexompata(ompatonvexsubsets
ofloallyonvextopologiallinearspaes)andaÆnemaps. LetU :Conv!Comp
be the forgetful funtor. A funtor F is alled onvex if there exists afuntor
F 0
: Comp ! Conv suh that F =UF 0
. It is easy to see that the funtors V,
E, EA, O and P are onvex. It is shown in [14℄ that for eah onvexfuntor
F there exists a unique natural transformation l : P ! F suh that the map
lX:PX !FX isanaÆneembeddingforeahompatumX.
Lemma2. P
Y =(lX) 1
(F
Y).
Proof: Take any measure 2 P(X) suh that lX() = 0
2 F
Y. By the
denition of F
Y it means that 0
2 FB for some ompatum B Y. We
will show that 2 PB P
Y. Choose a ompat absolute retrat T whih
obtained by attahing X and T by B. By r : Z ! T denote the retration.
Sinelisanaturaltransformationand ristheidentityonT Z, wehavethat
F(r)ÆlZ()= 0
=lTÆP(r)(). Hene,=P(r)()2P(T)dueto injetivity
oflZ. Therefore,2PX\PT =PB. Thelemma isproved.
We need some notions from innite-dimensional topology. See [4℄ for more
details. All spaesareassumed tobemetrizableand separable. A losedsubset
AofaompatumT isalledZ-setifthereexistsahomotopyH :T[0;1℄!T
suh that Hj
Tf0g
= id
Tf0g
and H(T (0;1℄)\A = ;; aountable unionof
Z-setsofT isalled aZ-set.
WedonotknowifF
Y isontainedinaZ-setofFXforanyonvexfuntorF.
Thus, weintroduesomeadditional property. We onsiderthe ompatumFX
asaonvexsubsetof aloallyonvexlinearspae.
ReallthatforanysubsetAofalinearspaeLthenotationa(A)standsfor
theaÆnehullofA,that is,theseta(A)=fta+(1 t)bja;b2A;t2Rg.
Denition2. AonvexfuntorF :Comp!Compisalledstronglyonvexiffor
eahompatumX,eahlosedsubsetAX wehave(FXnFA)\aFA=;.
Proposition 8. EahonvexsubfuntorF ofthefuntorV is stronglyonvex.
Proof: ByLemma 1anyelementfromaFAtakesthesamevalueatanytwo
funtionsfrom C(X)whihoinideonA,whihisnottrueforfuntionalsfrom
FXnFA.
Proposition 9. LetF beastronglyonvexfuntor. Then F
Y is ontainedin
aZ-setinFX.
Proof: Take any y 2 XnY. Then F
Y F
(Xnfyg), and Xnfyg an be
represented as a ountable union of its ompat subsets A
n
with the property
that A
n
int A
n+1
, hene, F
(Xnfyg) = S
n2N F(A
n
). Let us show that all
F(A
n
) are Z-sets in FX. Take any 2 FX nF
(X nfyg) and the set Z =
ft+(1 t)jt2(0;1℄;2F
(Xnfyg)g. SineF isstronglyonvex,wehave
Z\F
(Xnfyg)=;. SineZ isaonvexanddensesubsetofFX,thereexistsa
homotopyH :FX[0;1℄!FXsuhthatH(FX(0;1℄)Z(see,forexample,
Example12,13toSetion1.2 in[4℄).
Now,wearegoingto obtaintheompletetopologiallassiationofthepair
(FX;F
Y)whereXisametrizableompatumandY itsproperdenseG
Æ
-subset.
Weneedsomeharaterizationtheorems.
TheoremA ([8℄). LetCbeaninnite-dimensionaldenseonvexsubspaeof a
onvexmetrizableompatumK,CisontainedinaZ-setof Kandadditionally
letCbeaountableunionofitsnite-dimensionalompatsubspaes. Thenthe
pair(K ;C)is homeomorphito(Q;).
TheoremB ([7℄). LetKbeaonvexmetrizableompatum,andletCK be
itsproperdenseonvex-ompatsubspaethatontainsaninnite-dimensional
onvexompatumandis ontainedin aZ-set of K. Then thepair (K ;C)is
Thefollowingtheoremfollowsfrom5.3.6,5.2.6,3.1.10in[4℄.
TheoremC. LetKbeaonvexompatsubsetofaloallyonvexlinearmetri
spae, and let C K be its proper dense onvex F
Æ
subspae suh that C is
ontainedin aZ-set of K,(KnC)\aC=;,andadditionallythereexists a
ontinuousembeddingh:Q!K suhthat h 1
(C)=
!
. Thenthepair(K ;C)
ishomeomorphito thepair(Q;
!
).
Theorem2. LetF beastronglyonvexfuntor,X isametrizableompatum
andY isitsproperdenseG
Æ
-subset. Thepair(FX;F
Y)ishomeomorphito
(1) (Q;),if Y isadisretesubspaeof X andF(n)isnite-dimensionalfor
eah n2N;
(2) (Q;),if Y isadisretesubspaeof X andF(n)isinnite-dimensional
forsomen2N orY isaloallyompatnon-disretesubspaeof X;
(3) (Q;
!
),if Y isnotloally ompat.
Proof: Itiseasytoseethat F
Y isaonvexsubsetofFX.
Weprovethe rst assertion. Sine X is metrizable, Y is ountable. We an
represent Y = S
1
n=1 Y
n
where jY
n
j = n. Then F
Y =
S
1
n=1 FY
n
. Sine PY
n
ouldbeonsidered asan (n 1)-dimensionalsubspaeof FY
n
, the spae F
Y
isinnite-dimensional. Moreover,F
Y is aZ-set byProposition 9. Sineeah
FY
n
isanite-dimensionalompatum,weanapplyTheorem A.
We prove the seond assertion. In the ase when Y is disrete, FY
n is an
innite-dimensional onvex ompatumfor some n. When Y is not disrete, it
ontains an innite ompatum Y 0
and FY 0
is an innite-dimensional onvex
ompatum. WeapplyProposition9andTheoremB.
For the third assertion, note that the pair (PX;P
Y) is homeomorphi to
(Q;
!
)[2℄. Sine F isstronglyonvex,wehave(FXnF
Y)\aF
Y =;. We
applyLemma2,Proposition9andTheoremC.
Corollary 2. Suppose that F isastrongly onvexfuntor. Then foranysepa-
rablemetrizablespaeX
(1) X
=
N implies F
(X)
= Q
f
in ase F(n) is nite-dimensional for any
n2N orF
(X)
=
otherwise;
(2) if X isloallyompatnon-disreteandnon-ompatthenF
(X)
=
;
(3) if X istopologially ompletenotloally ompatthenF
(X)
=
!
.
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DepartmentofMehanisandMathematis,LvivNationalUniversity,
Universytetskast. 1,79000Lviv, Ukraine
E-mail: razymathsukr.net
InstituteofMathematis,CasimirustheGreatUniversity,Bydgoszz,
Poland
and
DepartmentofMehanisandMathematis,LvivNationalUniversity,
Universytetskast. 1,79000Lviv, Ukraine
E-mail: tarasradulyahoo.o.uk
(Reeived June4,2011 , revised February2,2012)
